hsg tim gia tri lon nhatnho nhat

7/28/2019 HSG Tim Gia Tri Lon Nhatnho Nhat

1/28

A. Cc kin thc thng s dng l:

+ Bt ng thc Csi: Cho hai s khng m a, b; ta c bt ng thc:

2

a bab

+ ;

Du = xy ra khi v ch khi a = b.

+ Bt ng thc: ( ) ( ) ( )2 2 2 2 2ac bd a b c d + + + (BT: Bunhiacopxki);

Du = xy ra khi v ch khia b

c d= .

+ a b a b+ + ; Du = xy ra khi v ch khi ab 0.

+ S dng bnh phng tm gi tr ln nht, gi tr nh nht.

Nu [ ]2

( )y a f x= + th min y = a khi f(x) = 0.

Nu [ ] 2( )y a f x= th max y = a khi f(x) = 0.

+ Phng php tm min gi tr (cch 2 v d 1 dng 2).

C. CC DNG TON V CCH GII

Dng 1 :CC BI TON M BIU THC CHO L MT A THC

Bi ton 1:Tm GTNN ca cc biu thc:

a) 24 4 11A x x= + +

b) B = (x-1)(x+2)(x+3)(x+6)c) 2 22 4 7C x x y y= + +

Gii:

a) ( )22 24 4 11 4 4 1 10 2 1 10 10A x x x x x= + + = + + + = + +

Min A = 10 khi1

2x = .

b) B = (x-1)(x+2)(x+3)(x+6) = (x-1)(x+6)(x+2)(x+3)

= (x2

+ 5x 6)(x2

+ 5x + 6) = (x2

+ 5x)2

36 -36 Min B = -36 khi x = 0 hoc x = -5.

c) 2 22 4 7C x x y y= + +

= (x2 2x + 1) + (y2 4y + 4) + 2 = (x 1)2 + (y 2)2 + 2 2

Min C = 2 khi x = 1; y = 2.

1


2/28

Bi ton 2:Tm GTLN ca cc biu thc:

a) A = 5 8x x2

b) B = 5 x2 + 2x 4y2 4y

Gii:

a) A = 5 8x x2 = -(x2 + 8x + 16) + 21 = -(x + 4)2 + 21 21 Max A = 21 khi x = -4.

b) B = 5 x2 + 2x 4y2 4y

= -(x2 2x + 1) (4y2 + 4y + 1) + 7

= -(x 1)2 (2y + 1)2 + 7 7

Max B = 7 khi x = 1,1

2y = .

Bi ton 3:Tm GTNN ca:

a) 1 2 3 4M x x x x= + + +

b) ( )2

2 1 3 2 1 2N x x= +

Gii:

a) 1 2 3 4M x x x x= + + +

Ta c: 1 4 1 4 1 4 3x x x x x x + = + + =

Du = xy ra khi v ch khi (x 1)(4 x) 0 hay 1 4x

2 3 2 3 2 3 1x x x x x x + = + + =

Du = xy ra khi v ch khi (x 2)(3 x) 0 hay 2 3x

Vy Min M = 3 + 1 = 4 khi 2 3x .

b) ( )22

2 1 3 2 1 2 2 1 3 2 1 2N x x x x= + = +

t 2 1t x= th t 0

Do N = t2 3t + 2 = 232 1( ) 4t 14

N .

Du = xy ra khi v ch khi3 3

02 2

t t = =

Do 1

4N= khi

3 52 1

3 3 2 42 13 12 2

2 12 4

x xt x

x x

= = = =

= =

2


3/28

Vy min1 5

4 4N x= = hay

1

4x = .

Bi ton 4:Cho x + y = 1.Tm GTNN ca biu thc M = x3 + y3.

Gii:

M = x3 + y3 = (x + y)(x2 xy + y2) = x2 - xy + y2

22 2 2 2

2 21 ( )2 2 2 2 2 2 2

x y x y x yxy x y

= + + + = + +

2 21 ( )2

M x y +

Ngoi ra: x + y = 1 x2 + y2 + 2xy = 1 2(x2 + y2) (x y)2 = 1

=> 2(x2 + y2) 1

Do 2 2 1

2x y+ v2 2 1 1

2 2x y x y+ = = =

Ta c: 2 21

( )2

M x y + v 2 21 1 1 1

( ) .2 2 2 4

x y M+ =

Do 1

4M v du = xy ra

1

2x y = =

Vy GTNN ca1 1

4 2M x y= = =

Bi ton 5: Cho hai s x, y tha mn iu kin:

(x

2

y

2

+ 1)

2

+ 4x

2

y

2

x

2

y

2

= 0.Tm GTLN v GTNN ca biu thc x2 + y2.

Gii:

(x2 y2 + 1)2 + 4x2y2 x2 y2 = 0

[(x2 + 1) y2]2 + 4x2y2 x2 y2 = 0

x4 + 2x2 + 1 + y4 2y2(x2 + 1) + 4x2y2 x2 y2 = 0

x4 + y4 + 2x2y2 + x2 3y2 + 1 = 0

x4 + y4 + 2x2y2 - 3x2 3y2 + 1 = -4x2

(x2+y2)2-3(x2+y2)+1=-4x2

t t = x2 + y2. Ta c: t2 3t + 1 = -4x2

Suy ra: t2 3t + 1 0

3


4/28

2

2

3 9 52. . 0

2 4 4

3 5 3 5

2 4 2 2

5 3 5

2 2 2

3 5 3 52 2

t t

t t

t

t

+

+

V t = x2 + y2 nn :

GTLN ca x2 + y2 = 3 52

+

GTNN ca x2 + y2 = 3 52

Bi ton 6: Cho 0 a, b, c 1. Tm GTLN v GTNN ca biu thc:

P = a + b + c ab bc ca.

Gii:

Ta c: P = a + b + c ab bc ca

= (a ab) + (b - bc) + (c ca)

= a(1 b) + b(1 c) + c(1 a) 0 (v 0 , , 1a b c )

Du = c th xy ra chng hn: a = b = c = 0

Vy GTNN ca P = 0Theo gi thit ta c: 1 a 0; 1 b 0; 1 c 0;

(1-a)(1-b)(1-c) = 1 + ab + bc + ca a b c abc 0

P = a + b + c ab bc ac 1 1abc

Du = c th xy ra chng hn: a = 1; b = 0; c ty [ ]0;1

Vy GTLN ca P = 1.

Bi ton 7: Cho hai s thc x, y tha mn iu kin: x2 + y2 = 1.

Tm GTLN v GTNN ca x + y.

Gii:

Ta c: (x + y)2 + (x y)2 (x + y)2

2(x2 + y2) (x + y)2

M x2 + y2 = 1 (x + y)2 2

4


5/28

2 2 2x y x y + +

- Xt 2x y+

Du = xy ra2

22

x yx y

x y

= = =+ =

- Xt2x y

+ Du = xy ra

2

22

x yx y

x y

= = =+ =

Vy x + y t GTNN l 2 2

2x y

= = .

Bi ton 8: Cho cc s thc dng tha mn iu kin: x2 + y2 + z2 27.

Tm GTLN v GTNN ca biu thc: x + y + z + xy + yz + zx.

Gii:

Ta c: (x y)2 + (x z)2 + (y z)2 0 2x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx 0

(x + y + z)2= x2 + y2 + z2 +2(xy + yz + zx) 3(x2 + y2 + z2) 81

x + y + z 9 (1)

M xy + yz + zx x2 + y2 + z2 27 (2)

T (1) v (2) => x + y + z + xy + yz + zx 36.

Vy max P = 36 khi x = y = z = 3.

t A = x + y + z v B = x2 + y2 + z22 2( 1) 1 1

2 2 2 2

A B A B BP A

+ + + = + =

V B 27 1

2

B + -14 P -14

Vy min P = -14 khi 2 2 21

27

x y z

x y z

+ + =

+ + =

Hay 13; 13; 1x y z= = = .

Bi ton 9:Gi s x, y l cc s dng tha mn ng thc: x + y = 10 . Tm gi tr ca x v y

biu thc: P = (x4 + 1)(y4 + 1) t GTNN. Tm GTNN y.

Gii:

Ta c: P = (x4 + 1)(y4 + 1) = (x4 + y4) + (xy)4 + 1

t t = xy th:5


6/28

x2 + y2 = (x + y)2 2xy = 10 2t

x4 + y4 = (x2 + y2)2 2x2y2 = (10 2t)2 2t2 = 2t2 40t + 100

Do : P = 2t2 40t + 100 + t4 + 1 = t4 + 2t2 40t + 101

= (t4 8t2 + 16) + 10(t2 4t + 4) + 45 = (t2 4)2 + 10(t 2)2 + 45

45P v du = xy ra x + y = 10 v xy = 2.Vy GTNN ca P = 45 x + y = 10 v xy = 2.

Bi ton 10:

Cho x + y = 2. Tm GTNN ca biu thc: A = x2 + y2.

Gii:

Ta c: x + y = 2 y = 2 x

Do : A = x

2

+ y

2

= x

2

+ (2 x)

2

= x2 + 4 4x + x2

= 2x2 4x + 4

= 2( x2 2x) + 4

= 2(x 1)2 + 2 2

Vy GTNN ca A l 2 ti x = y = 1.

Dng 2 : CC BI TON M BIU THC CHO L MT PHN THC

Bi ton 1:

Tm GTLN v GTNN ca: 24 3

1

xy

x

+=

+.

Gii:

* Cch 1:2

2 2

4 3 ax 4 3

1 1

x x ay a

x x

+ + + = = +

+ +Ta cn tm a 2ax 4 3x a + + l bnh phng ca nh thc.

Ta phi c:1

' 4 (3 ) 04

aa a

a

= = + = =

- Vi a = -1 ta c:2 2

2 2

4 3 x 4 4 ( 2)1 1

1 1 1

x x xy

x x x

+ + + += = + = +

+ + +

6


7/28

1.y Du = xy ra khi x = -2.

Vy GTNN ca y = -1 khi x = -2.

- Vi a = 4 ta c:2 2

2 2

4 3 -4x 4 1 (2 1)4 4 4

1 1 1

x x xy

x x x

+ + = = + =

+ + +

Du = xy ra khi x = 12

.

Vy GTLN ca y = 4 khi x =1

2.

* Cch 2:

V x2 + 1 0 nn: 224 3

yx 4 3 01

xy x y

x

+= + =

+(1)

y l mt gi tr ca hm s (1) c nghim

- Nu y = 0 th (1)3

4x =

- Nu y 0 th (1) c nghim ' 4 ( 3) 0y y = ( 1)( 4) 0y y + 1 0

4 0

y

y

+

hoc1 0

4 0

y

y

+

1 4y

Vy GTNN ca y = -1 khi x = -2.

Vy GTLN ca y = 4 khi x =1

2

.

Bi ton 2: Tm GTLN v GTNN ca:2

2

1

1

x xA

x x

+=

+ +.

Gii:

Biu thc A nhn gi tr a khi v ch khi phng trnh n x sau y c nghim:2

2

1

1

x xa

x x

+=

+ +(1)

Do x2 + x + 1 = x2 + 2. 12 .x +2

1 3 1 3 04 4 2 4

x + = + +

Nn (1) ax2 + ax + a = x2 x + 1 (a 1)x2 + (a + 1)x + (a 1) = 0 (2)

Trng hp 1: Nu a = 1 th (2) c nghim x = 0.

Trng hp 2: Nu a 1 th (2) c nghim, iu kin cn v l 0 , tc

l:

7


8/28

2( 1) 4( 1)( 1) 0 ( 1 2 2)( 1 2 2) 0

1(3 1)( 3) 0 3( 1)

3

a a a a a a a

a a a a

+ + + + +

Vi1

3a = hoc a = 3 th nghim ca (2) l

( 1) 1

2( 1) 2(1 )

a ax

a a

+ += =

Vi1

3

a = th x = 1

Vi a = 3 th x = -1

Kt lun: gp c 2 trng hp 1 v 2, ta c:

GTNN ca1

3A = khi v ch khi x = 1

GTLN ca A = 3 khi v ch khi x = -1

Bi ton 3:

a) Cho a, b l cc s dng tha mn ab = 1. Tm GTNN ca biu thc:2 2 4( 1)( )A a b a ba b

= + + + ++

.

b) Cho m, n l cc s nguyn tha1 1 1

2 3m n+ = . Tm GTLN ca B = mn.

Gii:

a) Theo bt ng thc Csi cho hai s dng a2 v b2

2 2 2 22 2 2a b a b ab+ = = (v ab = 1)2 2 4 4 4

( 1)( ) 2( 1) 2 ( ) ( )A a b a b a b a b a ba b a b a b = + + + + + + + = + + + + ++ + +

Cng theo bt ng thc csi cho hai s dng a + b v4

a b+.

Ta c: (a + b) +4 4

2 ( ). 4a ba b a b

+ =+ +

Mt khc: 2 2a b ab+ =

Suy ra:4

2 ( ) ( ) 2 4 2 8A a b a ba b

+ + + + + + + =+

Vi a = b = 1 th A = 8Vy GTNN ca A l 8 khi a = b = 1.

b) V1 1 1

2 3m n+ = nn trong hai s m, n phi c t nht mt s dng. Nu c mt trong

hai s l m th B < 0. V ta tm GTLN ca B = mn nn ta ch xt trng hp c hai s

m, n cng dng.

8


9/28

Ta c:1 1 1

3(2 ) 2 (2 3)( 3) 92 3

m n mn m nm n

+ = + = =

V m, n N* nn n 3 -2 v 2m 3 -1.

Ta c: 9 =1.9 = 3.3 = 9.1; Do xy ra:

+2 3 1 2

3 9 12

m m

n n

= =

= =

v B = mn = 2.12 = 24

+2 3 1 3

3 3 6

m m

n n

= = = =

v B = mn = 3.6 = 18

+2 3 9 6

3 1 4

m m

n n

= = = =

v B = mn = 6.4 = 24

Vy GTLN ca B = 24 khi2

12

m

n

= =

hay6

4

m

n

= =

Bi ton 4: Gi s x v y l hai s tha mn x > y v xy = 1. Tm GTNN ca biu

thc:

2 2x yA x y+= .

Gii:

Ta c th vit:2 2 2 2 22 2 ( ) 2x y x xy y xy x y xy

Ax y x y x y

+ + + += = =

Do x > y v xy = 1 nn:2( ) 2 2 2

2 2

x y xy x y x yA x y

x y x y x y

+ = = + = + +

V x > y x y > 0 nn p dng bt ng thc csi vi 2 s khng m, ta c:

22. .2 2x y x yA x y +

Du = xy ra22 ( ) 4 ( ) 2

2

x yx y x y

x y

= = =

(Do x y > 0)

T :2

2 32

A + =

Vy GTNN ca A l 32

1

x y

xy

= =

1 2

1 2

x

y

= + = +

hay1 2

1 2

x

y

=

= Tha iu kin xy = 1

Bi ton 5: Tm GTLN ca hm s: 21

1y

x x=

+ +.

Gii:

Ta c th vit: 221 1

1 1 32 4

yx x

x

= =+ + + +

9


10/28

V2

1 3 3

2 4 4x + +

. Do ta c:

4

3y . Du = xy ra

1

2x = .

Vy: GTLN ca4

3y = ti

1

2x

=

Bi ton 6: Cho t > 0. Tm GTNN ca biu thc:1

( )4

f t tt

= + .

Gii:

Ta c th vit:2 2 21 4 1 (2 1) 4 (2 1)

( ) 14 4 4 4

t t t t f t t

t t t t

+ + = + = = = +

V t > 0 nn ta c: ( ) 1f t

Du = xy ra1

2 1 02

t t = =

Vy f(t) t GTNN l 1 ti1

2t= .

Bi ton 7: Tm GTNN ca biu thc:2

2

1( )

1

tg t

t

=

+.

Gii:

Ta c th vit:2

2 2

1 2( ) 1

1 1

tg t

t t

= =

+ +

g(t) t GTNN khi biu thc 22

1t +t GTLN. Ngha l t2 + 1 t GTNN

Ta c: t2 + 1 1 min (t2 + 1) = 1 ti t = 0 min g(t) = 1 2 = -1

Vy GTNN ca g(x) l -1 ti t = 0.

Bi ton 8: Cho x, y, z l cc s dng tha mn iu kin: xyz = 1. Tm GTNN ca

biu thc: 3 3 31 1 1

( ) ( ) ( )E

x y z y z x z x y= + +

+ + + .

Gii:

t 1 1 1 1; ; 1a b c abcx y z xyz

= = = = =

Do :1 1

( ). ( )a b x y a b xy x y c a bx y

+ = + + = + + = +

Tng t: y + z = a(b + c)

z + x = b(c + a)

10


11/28

3 3 3

2 2 23 3 3

1 1 1 1 1 1. . .( ) ( ) ( )

1 1 1. . .

( ) ( ) ( )

Ex y z y z x z x y

a b ca b c

a b c b c a c a b b c c a a b

= + ++ + +

= + + = + ++ + + + + +

Ta c:3

2

a b c

b c c a a b+ +

+ + +(1)

Tht vy: t b + c = x; c + a = y; a + b = z

2

; ;2 2 2

x y za b c

y z x z x y x y za b c

+ + + + =

+ + + = = =

Khi ,2 2 2

a b c y z x z x y x y zVT

b c c a a b x y z

+ + + = + + = + +

+ + +1 1 1 3 3 3

1 1 12 2 2 2 2 2

y x z x z y

x y x z y z

= + + + + + + + =

Nhn hai v (1) vi a + b + c > 0. Ta c:( ) ( ) ( ) 3

( )2

a a b c b a b c c a b ca b c

b c c a a b

+ + + + + ++ + + +

+ + +2 2 2 33 3 3

2 2 2 2

a b c a b c abcE

b c c a a b

+ + + + =

+ + +

GTNN ca E l3

2khi a = b = c = 1.

Bi ton 9: Cho x, y l cc s thc tha mn: 4x2 + y2 = 1 (*).Tm GTLN, GTNN ca biu thc:

2 3

2 2

x ya

x y

+=

+ + .

Gii:

T2 3

2 2

x ya

x y

+=

+ + a(2x+y+z) = 2x+3y

2ax + ay + 2a 2x +3y = 0

2(a 1)x + (a 3)y = -2a (1)

p dng bt ng thc Bunhiacopxki cho hai b s (2x; y) v (a 1; a 3)

Ta c: 4a2 = [2x(a-1)+y(a-3)]2 (4x2+y2).[(a-1)2+(a-3)2]

=> 2 2 24 ( 1) ( 3)a a a= + (v 4x2+y2 = 1)

Do ta c: 2 2 2 2 24 ( 1) ( 3) 2 1 6 9a a a a a a a + = + + +

2 22 8 10 0 4 5 0a a a a + +

11


12/28

5 0( 1)( 5) 0

1 0

aa a

a

+ +

(V a + 5 > a 1) 1 5a

* Thay a = 1 vo (1) ta c: -2y = -2 y = 1

Thay y = 1 vo (*) ta c: x = 0 (x; y) = (0;1)

* Thay a = -5 vo (1) ta c: 2(-5 1)x + (-5 3)y = -2(-5)6 512 8 10 6 4 5

4xx y x y y = + = =

Thay vo (*) ta c:2

2 6 54 14

xx

+ =

2 3 4100 60 9 010 5

x x x y + + = = = 3 4

( ; ) ;10 5

x y =

Vy GTLN ca a l 1 khi x = 0; y = 1.

GTNN ca a l -5 khi3 4

;

10 5

x y= = .

Bi ton 10:

Gi s x, y l hai s dng tha mn iu kin: x + y = 1.

Hy tm gi tr nh nht cu biu thc:

M =22

1 1x y

x y

+ + +

Gii:

Ta c: M =

22

1 1x yx y + + +

=2 2

2 2

1 12 2x y

x y+ + + + +

= 4 + x2 + y2 + ( )2 2

2 2

2 2 2 2

14 1

x yx y

x y x y

+= + + +

V x, y > 0 nn ta c th vit:

( )2

0 2x y x y xy +

M x + y = 1 nn 1 2 21 12 2 16xy x yxy (1)

Du = xy ra khi v ch khi1

2x y= =

Ngoi ra ta cng c:2 2 2 2 2 2 2( ) 0 2 2( ) 2x y x y xy x y xy x y + + + +

2 2 2 2 22( ) ( ) 2( ) 1x y x y x y + + + (v x + y = 1)12


13/28

2 2 1

2x y + (2)

Du = xy ra khi v ch khi1

2x y= =

T (1) v (2) cho ta:2 2

2 2

1 1 254 ( )(1 ) 4 (1 16)

2 2M x y

x y= + + + + + =

Do :25

2M

Du = xy ra khi v ch khi ng thi (1) v (2) cng xy ra du = ngha l khi1

2x y= =

Vy GTNN ca25

2M = khi v ch khi

1

2x y= = .

* Dng 3: CC BI TON M BIU THC CHO C CHA CN THC.

Bi ton 1: Tm GTLN ca hm s: 2 4y x x= + .

Gii:

* Cch 1:

iu kin:2 0

2 4(*)4 0

xx

x

p dng bt ng thc Bunhiacopxki: (ac + bd)

2

(a

2

+ b

2

)(c

2

+ d

2

)Du = xy ra khi v ch khi

a b

c d= .

Chn 2; 1; 4 ; 1a x c b x d = = = = vi 2 4x

Ta c:

( ) ( ) ( ) ( )

( ) ( )

2 2 22 2 2

2

2

2 4 2 4 . 1 1

2 4 .2

4 2

y x x x x

y x x

y y

= + + + +

V y > 0 nn ta c: 0 2y<

Du = xy ra 2 4 2 4 3x x x x x = = = (Tha mn (*))

Vy GTLN ca y l 2 ti x = 3.

* Cch 2:

Ta c: 2 4y x x= +

13


14/28

iu kin:2 0

2 44 0

xx

x

V y > 0 nn y t GTLN khi v ch khi y2 t GTLN.

Ta c: 2 22 4 2 ( 2)(4 ) 2 2 ( 2)(4 )y x x x x y x x= + + = +

Do2 0

2 44 0

xx

x

nn p dng bt ng thc csi cho hai s khng m

cho ta: 2 ( 2)(4 ) ( 2) (4 ) 2x x x x + =

Do 2 2 2 4y + =

Du = xy ra 2 4 3x x x = = (tha mn iu kin).

Vy GTLN ca hm s y l 2 ti x = 3.

Bi ton 2: Tm GTLN, GTNN ca hm s: 3 1 4 5 (1 5)y x x x= + .

Gii:

a) GTLN:

p dng bt ng thc Bunhiacopxki cho hai b s:

(3; 4) v ( ( 1; 5 )x x ta c:

( ) ( )2 2

2 2 2 2(3. 1 4. 5 ) (3 4 ). 1 5 100y x x x x = + + + =

2 100y

=> y 10

Du = xy ra x =61

25(tha mn iu kin)

Vy GTLN ca y l10 khi x =61

25

* b) Ga tr nh nht:

Ta c: y = 3 1 4 5 3 1 3 5 5x x x x x + = + +

= ( )3 1 5 5x x x + +

t: A = 1 5x x + th t2 = 4 + 2 ( ) ( )1 5x x 4

=> A 2 v du = xy ra khi x = 1 hoc x = 5

Vy y 3 . 2 + 0 = 6

14


15/28

Du = xy ra khi x = 5

Do GTNN ca y l 6 khi x = 5

Bi ton 3: GTNN ca y l 6 khi x = 5

Tm GTNN ca biu thc: M = ( )2 2

1994 ( 1995)x x + +

Gii:

M = ( )2 21994 ( 1995)x x + + = 1994 1995x x + +

p dng bt ng thc: a b a b+ + ta c:

M = 1994 1995 1994 1995x x x x + + = + +

=> M 1994 1995 1x x + =

Du = xy ra khi v ch khi (x 1994) . (1995 x) 0 1994 1995x

Vy GTNN ca M = 1 1994 1995x

Bi ton 4:

Tm GTNN ca B = 3a + 4 21 a vi -1 1a

Gii:

B = 3a + 4 ( )22 3 161 5 5 1

5 25a a a = +

V p dng bt ng thc C si vi hai s khng m cho ta

( )( )

2

2 2

2

3 1613 16 5 255 5 1 5 5

5 25 2 2

a aa a

+ + +

=> B2 29 25 41 25

5 52 25

a a + + =

=> Do B 5 v du = xy ra khi v ch khi.

2

35

161

25

a

a

= =

a =3

5

Vy GTNN ca B = 5 a =3

5

Bi ton 5:

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16/28

Tm GTNN ca biu thc:

A =2

3

2 2 7x x+ +

Gii:

iu kin: ( )2 22 7 0 2 1 8 0x x x x + + +

-(x-1)2 + 8 0 ( )2

1 8x

2 2 1 2 2x

1 2 2 2 2 1x +

Vi iu kin ny ta vit:

( )22 22 7 1 8 8 2 7 8 2 2x x x x x + = + => + =

=> 2 + ( )22 7 2 2 2 2 2 1x x + + = +

Do :

( )21 1 2 1

22 2 12 2 7x x

=

++ +

Vy A2 1

32

v du = xy ra x -1 = 0

x = 1 (tha mn iu kin)

Vy GTNN ca A = ( )3

2 1 12

x =

Bi ton 6:

Tm GTNN ca biu thc: A =2

5 3

1

x

x

Gii:

iu kin: 1 x2 > 0 x2 < 1 - 1 < x < 1

=> A > 0 => GTNN ca A A2 t GTNN.

Ta c: A

2

=

( )

( )

( )2 22

2 2 2

2

5 3 3 525 30 916 16

1 11

x xx x

x xx

+= = +

Vy GTNN ca A = 4 khi3

5x =

Bi ton 7: Cho x > 0 ; y = 0 tha mn x + y 1

16


17/28

Tm GTNN ca biu thc: A = 21x x

Gii:

iu kin: 1 x2 0 1 1x

p dng bt ng thc C si hai s: x2 0 v 1 x2 0Ta c: x2 + 1 x2 ( )2 2 22 1 1 2 1x x x x =>

11

22

A A =>

Vy GTLN ca A =1

2khi x = 2

2 hay x = 2

2

Bi ton 8:

Tm GTLN ca biu thc: y = 1996 1998x x +

Gii:

Biu thc c ngha khi 1996 1998x

V y 0 vi mi x tha mn iu kin 1996 1998x

p dng bt ng thc C si ta c:

2 ( ) ( )1996 1998 ( 1996) (1998 ) 2x x x x + =

Du = xy ra khi v ch khi x 1996 = 1998 x

x = 1997

Do y2 4 2y =>

Vy GTLN ca y l 2 khi x = 1997

Bi ton 9:

Cho 0 1x . Tm GTLN ca biu thc y = x + ( )2 1 x

Gii:

Ta c: ( )2 1y x x= + = x + 2 ( )1

12

x

V 0 1x nn 1 x 0

p dng bt ng thc C si i vi 2 s:1

2v (1 x) cho ta:

17


18/28

( ) ( )1 1 3

2 1 12 2 2

y x x x x= + + + =

Du = xy ra 1 1

12 2

x x= => =

Vy GTLN ca y l3

2ti x =

1

2

Bi ton 10:

Cho M = 3 4 1 15 8 1a a a a+ + + Tm TGNN ca M

Gii:

M = 3 4 1 15 8 1a a a a+ + +

= 1 4 1 4 1 8 1 16a a a a + + +

= ( ) ( )2 2

1 2 1 4a a + iu kin M xc nh l a 1 0 1a

Ta c: 1 2 1 4M a a= +

t x = 1a iu kin x 0

Do : M = 2 4x x +

Ta xt ba trng hp sau:1) Khi x 2 th ( )2 2 2x x x = =

V ( )4 4 4x x x = =

=> M = 2 x + 4 x = 6 2x 6 2.2 2 =Vy x < 2 th M 2

2) Khi x 4 th 2 2x x = v x-4 =x-4

=> M = 2 4 2 6 2 4 6 2x x x + = =

Vy x > 4 th M 2

3) Khi 2 < x < 4 th 2 2x x = v 4 4x x =

=> M = x 2 + 4 x = 2 (khng ph thuc vo x)

Trong trng hp ny th: 2 1 4a 0. Phng trnh cho c nghim vi mi m theo h thc Vi-t,

ta c:2 2 2 2 2

1 2 1 2 1 2( ) 2 (2 1) 2( 2) 4 6 5x x x x x x m m m m+ = + = = +

=2

3 11 112

2 4 4m +

=> Min ( ( )2 21 211

4x x+ = vi m =

3

4

Bi ton 3:Cho x, y l hai s tha mn: x + 2y = 3. Tm GTNN ca E = x2 + 2y2

Gi :

Rt x theo y v th vo E

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20/28

Bi ton 4:

Tm GTLN v GTNN ca biu thc: A = x2 + y2

Bit rng x v y l cc s thc tha mn: x2 + y2 xy = 4

Gi :

T x2 + y2 xy = 4 2x2 + 2y2 2xy = 8 A + (x y)2 = 8

Max A = 8 khi x = y

Mt khc: 2x2 + 2y2 = 8 + 2xy

3A = 8 + (x + y)2 8

=> A8

3 => min A =

8

3khi x = - y

Bi ton 5:Cho x, y tha mn: x2 + 4y2 = 25.

Tm GTLN v GTNN ca biu thc: M = x + 2y.

Gii:

p dng bt ng thc: Bunhiacpxki

(x +2y)2 2 2( 4 )x y + (12 + 12) = 50

2 50 50 50x y M+

Vy Max M = 50 khi x =5 5

;2 2 2

y =

Min M = -5 2 khi x = -5

2; y = -

5

2 2

Bi tan 6:

Cho x, y l hai s dng tha mn iu kin: xy = 1. Tm GTLN ca biu thc:A = 4 2 2 4

x y

x y x y+

+ +

Gi :

T (x2 y)2 4 2 20 2x y x y => +

=> 4 2 21

2 2

x x

x y x y =

+

20


21/28

Tng t: 4 21

2

y

y x

+

=> A 1 => Max A = 1 khi

2

2 1

1

x y

y x x y

xy

=

= = = =

Bi tan 7:

Tm GTNN ca biu thc:

A = ( ) ( )2 1 1 2 1 1x x x x+ + + + + +

Gi :

B = 1 1 1 1x x+ + + + => Min B = 2 khi - 1 0x

Bi ton 8: Tm GTNN ca biu thc:

B = (x a )2 + (x b)2 + (x c)2 vi a, b, c cho trc.

Gi :

Biu din B = ( )( )

22

2 2 23.3 3

a b ca b cx a b c

+ ++ + + + +

=> GTNN ca B = (a2 + b2 + c2) - ( )2

3

a b c+ +

Bi ton 9: Tm GTNN ca biu thc:

P = x2 2xy + 6y2 12x +3y + 45

Gi :

Biu din P = (x 6 y)2 + 5(y 1)2 + 4

Vy Min P = 4 khi y = 1 ; x = 7

Bi ton 10: Tm GTLN ca biu thc:

E = x2 + 2xy 4y2 + 2x + 10y 3

Gi :

Biu din E = 10 (x y 1)2 3 (y 2)2

21


22/28

=> GTLN ca E = 10 y = 2 ; x = 3

Bi ton 11: Tm GTLN ca biu thc: P = 2 4 5x y z+ +

Bit x, y, z l cc bin tha mn : x2 + y2 + z2 = 169

Gi : p dng bt ng thc Bunhiacpxki

Max P = 65 khi

=

=

=

==

5

513

5

52

5

26

542

z

y

x

zyx

Bi ton 12:

Tm GTNN ca biu thc sau:

a) A =2 1

2

x

x

++

b) B = 28

3 2x

+

c) C =2

2

1

1

x

x

+

Gi :

a) p dng bt ng thc C si cho ta:A = (x + 2) +

54 2 5 4

2x

+

b) B = 28

43 2x

+(v 2

1 1)

3 2 2x

+

c) C =2

2

21 1

1

x

x + =>

+Min C = - 1 khi x = 0

22

Vi x 0

Vi mi x

Vi mi x


23/28

Bi ton 13:

Tm GTNN ca biu thc A =2

2

2 2000;( 0)

x xx

x

+

Gi :

A =2 2 2 2

2 2

2000 2 2000 2000 ( 2000) 1999

2000 2000

x x x x

x x

+ +=

=2

2

( 2000) 1999 1999

2000 2000 2000

x

x

+

Vy Min A =1999

2000Khi x = 2000

Bi ton 14:

Tm GTNN ca biu thc:

P =4 3 2

2

4 16 56 80 356

2 5

x x x x

x x

+ + + ++ +

Gi :

Biu din P = 4 2 2256

( 2 5) 642 5

x xx x

+ + + + +

(p dng BT Csi)

=> Min P = 64 khi x = 1 hoc x = -3

Bi ton 15:

Tm GTNN ca A =2 4 4x x

x

+ +vi x > 0

B =

2

1

x

x vi x > 1

C =2

2

2

1

x x

x x

+ +

+ +

D =1

(1 ) 1xx

+ +

vi x > 0

E =5

1

x

x x+

vi 0 < x < 1

F =2

2 1

x

x+

vi x > 1

Gi :

A = x+4 4

4 2 4 8xx x

+ + = (v x > 0)

=> Min A = 8 khi x = 2

B =2 1 1 1

2 ( 1) 2 2 41 1

xx

x x

+= + + + =

(v x > 1)

23


24/28

=> Min B = 4 x = 2

C =2 2

2 2

( 1) 1 2 12

1 1

x x x x

x x x x

+ + + + + =

+ + + +

D = (1 + x)1 1

1 2 .2. 4xx x

+ =

(v x > 0)

E = ( ) ( )5 1 5 15 5 5 5 2 5 2 5 51 1 1x xx x x x x

x x x x x x ++ = + + + = +

F =1 1 2 1 2 1 1 2 1

22 1 2 1 2 2 1 2

x x x

x x x

+ + = + + +

=1 3

22 2

+ = => Min F =3

2khi x = 3.

Bi 16: Tm GTLN v GTNN ca biu thc:

P =2

2 2

8 6x xy

x y

++

Gi :

P = 9 -2

2 2

( 3 )1 1

y x

x y

+

+

P = 9 -2

2 2

( 3 )9

x y

x y

+

Bi 17: Cho x, y l hai s dng tha mn: x + y = 10Tm GTNN ca biu thc S =

1 1

x y+

Gi : S = yx 11 + =10

(10 )

x y

xy x x

+=

S c GTNN x(10-x) c GTLN x = 5.

=> GTNN ca S =2

5khi x = y = 5.

Bi 18: Tm GTNN ca biu thc:

E = 2 21 1x x x x+ + + +

Gi :

Ta c E > 0 vi mi x

Xt E2 = 2 (x2 + 1 + 4 2 1) 4x x+ +

24


25/28

=> Min E = 2 khi x = 0

Bi 19: Cho a v b l hai s tha mn: a 3 ; a + b 5

Tm GTNN ca biu thc S = a2 + b2

Gi :a+ b 5 2 2 10 3 2 13a b a b => + => + (v a 3)

=> 132 ( ) ( )2 2 23 2 13a b a b + +

=> Min S = 13

Bi 20:

Cho phng trnh: x

2

- 2mx 3m

2

+ 4m 2 = 0Tm m cho 1 2x x t GTNN.

Gi :' 2(2 1) 1 0m = + > => phng trnh lun c 2 nghim phn bit x1; x2. Theo

nh l vi-t ta c:

1 2

2

1 2

2

. 3 4 2

x x m

x x m m

+ =

= +

Do ( )2

1 2 4 2 4 4 2x x m = + = m RGTNN ca 1 2x x l 2 khi m =

1

2

Bi 21:

Tm gi tr nh nht ca:

y = 1 2 ... 1998x x x + + +

Gi :

y = ( ) ( )1 1 1998 2 1997x x x x + + +

+ + ( )998 999x x +

Ta c: 1 1998x x + nh nht bng 1997 khi x [ ]1;1998

2 1997x x + nh nht bng 1995 khi x [ ]2;1997

998 1999x x + nh nht bng 1 khi x [ ]999;1000

Vy y t GTNN bng 1 + 3 + + 1997

S cc s hng ca 1 + 3 + + 1997 l (1997 1) : 2 + 1 = 999

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26/28

Vy Min y = 9992 khi 999 1000x

Bi 22:

Cho biu thc: M = x2 + y2 + 2z2 + t2

Vi x, y, z, t l cc s nguyn khng m , tm gia str nh nht ca M v cc gi trtng ng ca x, y, z, t. Bit rng:

2 2 2

2 2 2

21

3 4 101

x y t

x y z

+ =

+ + =

Gi :

Theo gi thit: x2 y2 + t2 = 21

x2 + 3y2 + 4z2 = 101

=> 2x2 + 2y2 + 4z2 + t2 = 122

=> 2M = 122 + t2

Do 2M 122 61M

Vy Min M = 61 khi t = 0

T (1) => x > y 0 0x y x y => +

Do : (x + y )(x y) = 21.1 = 7.3

T (2) => 3y22

101 33 0 5y y => => Ta chn x = 5 ; y = 2 => z = 4

Vy Min M = 61 ti x = 5 ; y = 2 ; z = 4; t = 0

Bi 23:

Cho phng trnh: x4 + 2x2 +2ax (a 1)2 = 0 (1)

Tm gi tr ca a nghim ca phng trnh :

a) t GTNN.

b) t ga tr ln nht.

Gi :

Gi m l nghim ca phng trnh (1) th:

m4 + 2m2 + 2am + a2 + 2a + 1 = 0 (2)

26

(1)

(2)


27/28

Vit (2) di dng phng trnh bc hai n a.

a2 + 2 (m + 1) a + (m4 + 2m2 + 1) = 0

tn ti a th ' 0

Gii iu kin ny c m4 - m2 0 m(m 1) 0 0 1m

Vy nghm ca phng trnh t GTNN l 0 vi a = -1Vy nghm ca phng trnh t GTLN l 1 vi a = -2

Bi 24: Tm GTNN, GTLN ca t =2

2

2 2

1

x x

x

+ ++

Gi : V x2 + 1 > 0 vi mi x

t a =2

2

2 2

1

x x

x

+ ++

=> (a 1) x2 2 x +a 2 = 0 (1)

a l mt gi tr ca hm s (1) c nghim.

- Nu a = 1 th (1) x = 12

- Nu a 1 th (1) c nghim ' 0

Min A = 3 52

vi x = 1 5 3+ 5; ax A =2 2

M vi x = 5 1

2

Bi 25:

Tm GTNN, GTLN ca A =2 2

2 2

x xy y

x xy y

++ +

Gi : Vit A di dng sau vi y 0

(

2

2

2 2

11

11

x xy y a a

Aa ax x

y y

+ + = =

+ + + +

(tx

ay

= )

Gii tng t bi 24 c:1

33

A

Cn vi y = 0 th A = 1

Do : Min A = 13

vi x = y ; max A = 3 vi x = - y

Bi 26: Cho a + b = 1. Tm GTNN ca biu thc:

Q = a3 + b3 + ab

Gi :

Vi Q di dng Q = (a + b) ( )2

3a b ab ab + +

27


28/28

= 1 2ab = 1 2a (1 a)

=> Q = 2a2 2a + 11

2

Do : Min Q =1

2khi a = b =

1

2

hsg tim gia tri lon nhatnho nhat

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